3. Consider a pilot study in which n = 15 children enrolled in special education classes were randomly selected and tested for a certain type of learning disability. In the pilot study, y = 2 children tested positive for the disability. (a) Using a uniform prior distribution, nd the posterior distribution of 3, the fraction of stu dents who have the disability. Plot the posterior and nd a 95% condence interval. Researchers would like to recruit students with the disability to participate in a longterm study, but rst they need to make sure they can recruit enough students. Let f1 = 278 be the number of children in special education classes in this particular school district. and let 17 be the number of students with the disability. (b) (C) (d) Find Pr' = lY = 2), the posterior predictive distribution of 1}, by two methods: i. -iv -iayes' rule, _ = g.) _ fol my = magma? = mammal; Pr Y = 2) IS my = 2|9)p(a)da Pd? my 2] where 10(3) is the uniform prior density. 30(9) = 1 for ll